Polyhedron Coloring

Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron (Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one way to color a cube (Ball and Coxeter 1987, pp. 238-239) and 144 ways to color an icosahedron (Ball and Coxeter 1987, pp. 239-242). Given four colors, there are two distinct ways to color a tetrahedron (Ball and Coxeter 1987, p. 238) and four ways to color a dodecahedron, consisting of two enantiomorphous ways (Steinhaus 1999, pp. 196-198; Ball and Coxeter 1987, p. 238). Given five colors, there are four ways to color an icosahedron. Given six colors, there are 30 ways to color a cube (Steinhaus 1999, p. 167). These values are related to the chromatic polynomial of the corresponding dual skeleton graph, which however overcounts since it does not take rotational equivalence of colorings in the original solid into account.

The following table gives the numbers of ways to color faces of various solids using at most  colors (with no restriction about colors on adjacent faces). This can be computed by finding the graph automorphisms of the skeleton of the polyhedron, removing the symmetries that invert a face (leaving pure rotational symmetries only), then finding the induced symmetry group for the faces and applying the Pólya enumeration theorem.

solid	polynomial	OEIS	colorings for , 2, ...
cube		A047780	1, 10, 57, 240, 800, 2226, 5390, ...
dodecahedron		A000545	1, 96, 9099, 280832, 4073375, 36292320, ...
icosahedron		A054472	1, 17824, 58130055, 18325477888, 1589459765875, ...
octahedron		A000543	1, 23, 333, 2916, 16725, 70911, 241913, ...
tetrahedron		A006008	1, 5, 15, 36, 75, 141, 245, 400, 621, ...

REFERENCES

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 238-242, 1987.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 82-83, 1989.

Sloane, N. J. A. Sequences A000543, A000545, A006008/M3854, A047780/M4716, and A054472 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.